A Kobayashi and Bergman complete domain without bounded representations
aa r X i v : . [ m a t h . C V ] M a r A KOBAYASHI AND BERGMAN COMPLETE DOMAINWITHOUT BOUNDED REPRESENTATIONS
NIKOLAY SHCHERBINA AND LIYOU ZHANG
Abstract.
We construct an unbounded strictly pseudoconvex Kobayashi hy-perbolic and complete domain in C , which also possesses complete Bergmanmetric, but has no nonconstant bounded holomorphic functions. A question of existence of the (complete) Bergman metric on a given complexmanifold X is one of the important problems of complex analysis. When X = Ωis a bounded domain in C n , it is well-known that the Bergman metric on Ω exists.If Ω is further assumed to be bounded pseudoconvex with C -smooth boundary,then the Bergman completeness of Ω was proved by Ohsawa in [26] using the cel-ebrated Kobayshi’s criterion [24] (Here the Bergman completeness means that Ωis a complete metric space with respect to the Bergman metric). Later on, severalarticles appeared concerning the Bergman completeness for bounded pseudoconvexdomains in C n (see [10, 31] and references therein). Among them, it is worth tomention the papers of Blocki-Pflug [5] and Herbort [22] were independently a long-standing question of Bergman completeness for any bounded hyperconvex domainhas been solved (by a hyperconvex domain we mean here that it has a boundedcontinuous plurisubharmonic exhaustion function). A few years later this resultwas further generalized by Chen [6] to the case of hyperconvex manifolds.When X is an unbounded domain, or, more generally, a complex manifold,then not so much is known for the existence of the (complete) Bergman metricexcept the early work of Greene-Wu [18], which asserts that a simply-connectedcomplex manifold possesses a complete Bergman metric if it carries a completeK¨ahler metric with holomorphic sectional curvature which is negatively pinched.It is a bit surprising that such lower negative bound was dropped by Chen andZhang in [9], where the main ingredient used in the proof is the pluricomplexGreen function (see Section 4 below for the definition) and the L -method for the¯ ∂ -equation. In particular, they proved that a Stein manifold X possesses a Bergmanmetric, provided that X carries a bounded continuous strictly plurisubharmonicfunction. Recently, a new characterization for the existence of the Bergman metricon unbounded domains was given by Gallagher, Harz and Herbort in [14]. It wasproved there that a pseudoconvex domain with empty core (see Section 3.1 belowfor the definition) possesses a Bergman metric.Some other conditions (for certain unbounded X ) which are sufficient for pos-sessing a (complete) Bergman metric are also scattered in the literatures, see forexamples [2], [8], [27], [29] et al.A complex manifold X is called (Kobayashi) hyperbolic if its Kobayashi pseu-dodistance κ X is a distance (see Section 5 below for definitions). We say that X is complete hyperbolic, if ( X, κ X ) is a complete metric space. For example, all bounded domains in C n are hyperbolic, while complex manifolds containing entirecurves are not. The question of hyperbolicity has been intensively studied in thecase of compact complex manifolds. Nevertheless, there are many interesting andquite long standing conjectures in that context which are still open in their com-plete form, such as the Kobayashi conjecture and Green-Griffiths-Lang conjecture[12].In the case of open complex manifolds, the questions related to the Kobayashi hy-perbolicity are rather different. There are various characterizations of the hyperbol-icity. For examples, an analytic description due to Sibony [30] says that a complexmanifold X carrying a bounded strictly plurisubharmonic function is Kobayashi hy-perbolic. In [1], Abate proved that a complex manifold X is Kobayashi hyperbolicif and only if the space of holomorphic maps from the unit disc ∆ to X is relativelycompact (with respect to the compact-open topology) in the space of continuousmaps from ∆ into the one point compactification X ∗ of X . Later, Gaussier [15]gave sufficient conditions for hyperbolicity of an unbounded domain in terms ofthe existence of peak and antipeak functions at infinity. In [25], Nikolov and Pflugfound some conditions at infinity which guarantee the hyperbolicity of unboundeddomains. They also obtained a characterization of hyperbolicity in terms of asymp-totic behavior of the Lempert function. Recently, Gaussier and Shcherbina [17] gavea new sufficient condition for Kobayashi hyperbolicity of unbounded domains in C n using a concept of strong antipeak plurisubharmonic function at infinity. For moreinformation on the progress in this direction (conditions which characterize (com-plete) Kobayashi hyperbolic (open) manifolds) we refer the reader to the surveypaper of Gaussier [16].In the present paper, in contrast to the situation treated by Chen in [6] andSibony in [30] (when the existence of a bounded continuous strictly plurisubhar-monic function was assumed), we consider both the Bergman completeness and theKobayashi hyperbolicity for unbounded domains in C n which have neither boundedsmooth strictly plurisubharmonic functions nor nonconstant bounded holomorphicfunctions. More precisely, we prove here the following result. Main Theorem.
There exists an unbounded strictly pseudoconvex domain A ⊂ C with smooth boundary which has the following properties:(1) A possesses a complete Bergman metric.(2) A is Kobayashi hyperbolic and complete.(3) A possesses neither nonconstant bounded holomorphic functions, nor con-tinuous bounded strictly plurisubharmonic functions.(4) A is not Carath´eodory hyperbolic.(5) The core c (Ω) of Ω is nonempty, but the core c ′ (Ω) is empty. (See Section 3.1 below for the definitions of the cores c (Ω) and c ′ (Ω)).The construction of A is motivated by [17], where a Kobayashi hyperbolic Modeldomain with a nonempty core has been constructed. Using a characterization ofthe Bergman space in terms of the core (see [14, Remark 7(b)]), we show the exis-tence of the Bergman metric on A . For the completeness of this metric, we applya criterion given by Chen [7, Theorem 1.1], which uses the asymptotic behavior atinfinity of the volumes of sublevel sets of the pluricomplex Green function. Thecompleteness of Kobayashi metric follows from a criterion by Nikolov and Pflug[25, Proposition 3.6], which identifies the Kobayashi completeness and the local KOBAYASHI AND BERGMAN COMPLETE DOMAIN 3
Kobayashi completeness in the case when there are no Cauchy sequences (w.r.t.Kobayashi metric) converging to infinity. For the nonexistence of bounded holo-morphic functions on A , we use an argument similar to the Liouville type theoremproved in [21, Theorom 2.2], which says that any continuous bounded plurisubhar-monic function defined in a neighborhood of a Wermer type set E ⊂ A is constanton E . 1. Construction of a special Wermer type set in C Let { a n } be the enumeration of points running through the set Z + i Z ⊂ C z as follows: (0 , → (1 , → (1 , → (0 , → ( − , → ( − , → ( − , − → (0 , − → (1 , − → (2 , − → (2 , → · · · . Let { ε n } be a decreasing sequence of positive numbers converging to zero veryfast that will be further specified later. Then, as in [19], for each n ∈ N , we considerthe set E n := (cid:8) ( z, w ) ∈ C : w = ε √ z − a + ... + ε n √ z − a n (cid:9) . (1)By definition, P nk =1 ε k √ z − a k is a multi-valued function that takes 2 n values ateach point z ∈ C (counted with multiplicities). Therefore, for each z ∈ C \ ( Z + i Z )we can locally choose single-valued functions w ( n )1 ( z ) , w ( n )2 ( z ) , · · · , w ( n )2 n ( z ) such that n X k =1 ε k √ z − a k = n w ( n ) k ( z ) : k = 1 , , ..., n o . For every n ∈ N , we define a function P n : C → C as P n ( z, w ) := (cid:16) w − w ( n )1 ( z ) (cid:17) (cid:16) w − w ( n )2 ( z ) (cid:17) · · · (cid:16) w − w ( n )2 n ( z ) (cid:17) . Then each P n is a well defined holomorphic polynomial in z and w (see for details[19]). Moreover, provided that { ε n } is decreasing to zero fast enough, the sets E n = { P n = 0 } converge to a nonempty unbounded connected closed set E ⊂ C ,where the convergence is understood with respect to the Hausdorff metric on eachcompact subset of C . More precisely, E = lim n →∞ E n = n b ∈ C z,w : ∃ b n ∈ E n , n = 1 , , ..., with b = lim n →∞ b n o . (2)Define φ n ( z, w ) := 12 n log | P n ( z, w ) | . Then φ n converges uniformly on compact subsets of C \ E to a pluriharmonicfunction φ : C \ E → R , and lim ( z,w ) → ( z ,w ) φ ( z, w ) = −∞ for every ( z , w ) ∈ E .In particular, φ has a unique extension to a plurisubharmonic function on thewhole of C (see, for instance, [11, Chapter I, 5.24]), and the set E = { φ = −∞} iscomplete pluripolar. 2. Construction of the domain A Consider first a plurisubharmonic function ϕ defined on C by ϕ ( z, w ) := φ ( z, w ) + ρ ( | Re( z ) | ) + ρ ( | Im( z ) | ) , N. SHCHERBINA AND L. ZHANG where ρ is the convex function constructed in the Section 2.2 of [17], and observe,that for each t > U t := { ( z, w ) ∈ C : ϕ ( z, w ) < t } coincides with the domain F d of Lemma 2 in [17] for d = e t . The structure ofthe domains F d , which were systematically studied in [17], will be essential forproving in Section 5 below the Kobayashi completeness of our domain A . Thecrucial technical tool for this proof is the following property established in [17]: Property ( F ) : For each d > there exists r := r ( d ) > such that the domain F d contains no holomorphic disks of radius r > r (the last part of the statementmeans, more precisely, that for every holomorphic map h : ∆ r (0) → F d such that k h ′ (0) k = 1 one has r ≤ r ). Let us now pick one of these domains, for example U − , and denote it (to simplifyour notations) by U , then U will be a neighborhood of the Wermer type set E in C which is defined by U := { ( z, w ) ∈ C : ϕ ( z, w ) < − } . Then we set ˜ ϕ := − log( − ϕ ) + ˜ ρ ( k ζ k ) on U , (3)where ˜ ρ : [0 , ∞ ) → [0 , ∞ ) is a function with the following properties:(i) ˜ ρ is smooth strictly increasing and convex,(ii) ˜ ρ ( t ) = t , when t ∈ [0 , t ], for some t > t →∞ ˜ ρ ′ ( t ) = ∞ .Here ζ = ( z, w ) ∈ C and k ζ k := | z | + | w | is the Euclidean norm. Observe that,in view of strict monotonicity and convexity of the functions t → − log( − t ) and ˜ ρ ,the function ˜ ϕ is strictly plurisubharmonic on U . Then, since ˜ ϕ > ∂U , andsince ˜ ϕ = −∞ on the set E , we conclude that, maybe after a small perturbation ofthe function ˜ ρ , the domain A := { x ∈ U : ˜ ϕ < − } . (4)will be a smoothly bounded strongly pseudoconvex neighborhood of E such that A ⊂ U .Note that, by Theorem 6.1 below, we know that any bounded from above con-tinuous plurisubharmonic function u in the neighborhood of E is constant on E .Therefore, the core c ( A ) of A contains E . Observe also that, for each constant c > u (cid:12)(cid:12) E , the sublevel set { ζ ∈ A : u ( ζ ) c } is not relatively compact in A . Thisshows that the domain A is not hyperconvex.Now we will study the existence of the Bergman metric as well as the Bergmancompleteness of A . Note first that, since ϕ < − U , the function − log( − ϕ ) iswell defined and, moreover, it is plurisubharmonic, since ϕ is plurisubharmonic. Astraightforward calculation yields KOBAYASHI AND BERGMAN COMPLETE DOMAIN 5 i∂ ¯ ∂ ˜ ϕ = i∂ ¯ ∂ϕ − ϕ + i∂ϕ ∧ ¯ ∂ϕϕ + ˜ ρ ′ · i∂ ¯ ∂ k ζ k + ˜ ρ ′′ · i∂ k ζ k ∧ ¯ ∂ k ζ k (5) > ˜ ρ ′ ( k ζ k ) · i∂ ¯ ∂ k ζ k . Thus one can make i∂ ¯ ∂ ˜ ϕ arbitrary large whenever k ζ k is large and ˜ ρ grows suffi-ciently fast. Moreover, we can also force the volume of A to be finite if ˜ ρ goes to+ ∞ fast enough. This insures that the holomorphic polynomials are L -integrable.In particular, the Bergman kernel of A is non-degenerated.3. Existence of the Bergman metric
A notion of the core c (Ω) of a domain Ω ⊂ C n (or, more general, of a domainin a complex manifold M ) was introduced and intensively studied in [20], [21] and[28]. It can be defined as follows. Definition 3.1.
Let M be a complex manifold and let Ω ⊂ M be a domain. Thenthe set c (Ω) := (cid:8) ζ ∈ Ω : every smooth plurisubharmonic function on Ω that isbounded from above fails to be strictly plurisubharmonic in ζ (cid:9) is called the core of Ω . Similar definition can also be given for plurisubharmonic functions from differentsmoothness classes.Later on in [14] a slightly larger class of functions defined on a given domainΩ ⊂ C n , was considered:PSH ′ (Ω) := { φ ∈ PSH(Ω) : φ
6≡ −∞ and ν ( φ, · ) ≡ } , where PSH(Ω) denotes the family of plurisubharmonic functions in Ω and ν ( φ, ζ )denotes the Lelong number of φ at ζ , i.e. ν ( φ, ζ ) := lim inf ζ → ζ φ ( ζ )log | ζ − ζ | . In order to formulate a general sufficient condition for the infinite dimensionalityof the Bergman space of Ω, they introduced the following notion of the core c ′ (Ω)of Ω (which is a slight modification of the notion of the core c (Ω) given above). Definition 3.2.
Let Ω be a domain in C n . Then the core c ′ (Ω) of Ω is defined by c ′ (Ω) := { ζ ∈ Ω : every φ ∈ PSH ′ (Ω) that is bounded fromabove fails to be strictly plurisubharmonic at ζ } . The following criterion for the existence of the Bergman metric on Ω, in termsof the core c ′ (Ω), has been formulated in [14, Remark 7 (b)]. Theorem 3.1.
Every pseudoconvex domain Ω ⊂ C n with the empty core c ′ (Ω) possesses a Bergman metric. N. SHCHERBINA AND L. ZHANG
If we now consider the domain A defined in (4) and observe that the function˜ ϕ defined by (3) is negative strictly plurisubharmonic on this domain and has zeroLelong numbers (this easily follows from the fact that h ( t ) := − (1 /t ) log( − t ) is anincreasing convex function in t and the fact that lim t →−∞ h ( t ) = 0, see for details[14, Remark 7 (c)]), then, applying Theorem 3.1 to the domain A on the place ofΩ, we conclude that the Bergman metric on A exists. Remark 1.
Since the restriction of i∂ ¯ ∂ϕ to each vertical line C z ,w := { ( z, w ) ∈ C : z = z } is spread over the Cantor set E z := E ∩ C z ,w , one can prove thatthe Lelong numbers of the strictly plurisubharmonic function ϕ + ˜ ρ ( k ζ k ) + C k ζ k , C >
0, will also be identically equal to zero. Hence, we can use this function insteadof the function ˜ ϕ in the definition of the domain A and in all the arguments lateron. But the proof of the fact that the Lelong numbers of ˜ ϕ are equal to zero ismore elementary, that is why we have decided to use it here.4. Completeness of the Bergman metric
The main argument which we will use to prove Bergman completeness of thedomain A (see the Case 2 below) follows closely the arguments presented in theproof of Theorem 1.2 in [7]. For the reader’s convenience we give them here indetails.First we recall the notion of the pluricomplex Green function on an open setΩ ⊂ C n with logarithmic pole at ζ ∈ Ω: g Ω ( ζ, ζ ) := sup { u ( ζ ) } , ζ ∈ Ω , where the supremum is taken over all negative plurisubharmonic functions u on Ωsuch that u ( ζ ) log k ζ − ζ k + O (1) near ζ .For each a >
0, set A Ω ( ζ , a ) := { ζ ∈ Ω : g Ω ( ζ, ζ ) − a } . The following criterion for the Bergman completeness is given in [7, Theorem 1.1].
Theorem 4.1.
If a Stein manifold Ω possesses the Bergman metric, then it isBergman complete provided the following condition is satisfied:For any infinite sequence of points { ζ k } in Ω , without accumulation points in Ω ,there are a subsequence { ζ k j } , a number a > and a continuous volume form dV on Ω , such that for any compact subset K ⊂ Ω , one has Z K ∩ A Ω ( ζ kj ,a ) dV → as j → ∞ . Using this criterion we will now prove that Bergman metric on A is complete.Indeed, let { ζ k } be an infinite sequence of points in A without accumulationpoints in A . Then we have two possibilities: Case 1. { ζ k } admits an accumulation point on ∂ A . In this case, in view of strict pseudoconvexity of A , a standard localizationargument shows that the Bergman metric is complete. This is definitely known forexperts, but for completeness of the presentation we include the sketch of the proofhere. KOBAYASHI AND BERGMAN COMPLETE DOMAIN 7
Let p ∈ ∂ A be an accumulation point of { ζ k } and let U be a neighborhood of p .Since A is strictly pseudoconvex and, hence, locally has a strictly plurisubharmonicdefining function, there is a neighborhood V ⋐ U of the point p and a constant a >
0, such that for every point ζ ∈ V , one has A A ( ζ , a ) ⊂ U ∩ A . We fixan arbitrary point ζ ∈ V and consider a function f ∈ O ( U ∩ A ) which has theproperties f ( ζ ) = 0 and k f k L ( U ∩ A ) = 1. Then we set α := ¯ ∂ ( χ ◦ g A ) · f, (6)where g A is the pluricomplex Green function of A with a pole at ζ (written in whatfollows as g for simplicity) and χ : ( −∞ , + ∞ ) → [0 ,
1] is a smooth cut-off functionsuch that χ ( t ) ≡ t − a and χ ( t ) ≡ t > − a . Observe that α is a closed(0 , A .Next we consider the functions ψ := − log( − g )(7)and ϕ := (2 n + 2) g. (8)Since g is a negative plurisubharmonic function on A , a direct calculation (see alsothe estimate (5) above with ˜ ρ ≡
0) easily shows that i∂g ∧ ¯ ∂g ≤ g i∂ ¯ ∂ψ. (9)Since, by (6), α = ¯ ∂ ( χ ◦ g ) · f = χ ′ ◦ g · ¯ ∂g · f , it follows from (9) that | α | = | χ ′ ◦ g | · i∂g ∧ ¯ ∂g · | f | ≤ | χ ′ ◦ g | · g · | f | · i∂ ¯ ∂ψ = H · i∂ ¯ ∂ψ, (10)where H := | χ ′ ◦ g | · g ·| f | . Then, by the celebrated Donnelly-Fefferman’s existencetheorem (see for example [4, Theorem 3.1]), and in view of (7) and (10), there exists u ∈ L loc ( A ) such that ¯ ∂u = α and the following estimate holds Z A | u | e − ϕ · Z A H · e − ϕ . (11)From the definition of H and the fact that k f k L ( U ∩ A ) = 1 we can conclude by (11)that Z A | u | e − ϕ Z A | χ ′ ◦ g | g | f | e − ϕ · C ( a ) · a Z A A ( ζ ,a ) \ A A ( ζ , a ) | f | ˜ C ( a ) , where C ( a ) and ˜ C ( a ) are some constants that depend on a only (for A , U and V being fixed). It follows now from the fact that g ( ζ, ζ ) − log | ζ − ζ | is locallybounded and, hence, in view of (8), from the fact that the integrability of | u | e − ϕ = | u | e − (2 n +2) g is the same as the integrability of | u | | ζ − ζ | − (2 n +2) , that u ( ζ ) = 0and ∂u∂ζ ( ζ ) = 0. Therefore, if we set F := ( χ ◦ g ) f − u, (12)we will get a function F ∈ O ( A ) such that F ( ζ ) = f ( ζ ) and ∂F∂ζ ( ζ ) = ∂f∂ζ ( ζ ) . (13)By the estimate (11), we have that k F k L ( A ) k ( χ ◦ g ) f k L ( A ) + k u k L ( A ) (1 + ˜ C ( a )) k f k L ( U ∩ A ) =: ˜˜ C. N. SHCHERBINA AND L. ZHANG
Recall that the Bergman metric is defined by ds A ( ζ, X ) := K − A ( ζ ) sup {| Xf | : f ∈ O ( A ) with f ( ζ ) = 0 and k f k L ( A ) } , where X is a nonzero tangent vector at ζ . If we assume that the function f achievesthe above supremum on U ∩ A , we will get by (12) a function F ∈ O ( A ) such that,in view of (13), for any ζ ∈ V ⋐ U the following estimate holds K A ( ζ ) · ds A ( ζ, X ) > ˜˜ C − | XF | = ˜˜ C − | Xf | = ˜˜ C − K U ∩ A ( ζ ) · ds U ∩ A ( ζ, X ) . We can conclude now from the trivial inequality K A ( ζ ) K U ∩ A ( ζ ) that ds A ( ζ, X ) > ˜˜ C − ds U ∩ A ( ζ, X ) , for all ζ ∈ V . This completes the proof of the localization property for the Bergmanmetric and shows the completeness of the Bergman metric at the finite points of ∂ A . Case 2. ζ k → ∞ as k → ∞ . Consider a smooth cut-off function χ on C such that χ ( ζ ) = 1, when k ζ k and χ ( ζ ) = 0, when k ζ k ≥ , and then for every δ > u δ,k ( ζ ) := δ ˜ ϕ ( ζ ) + χ ( ζ − ζ k ) log k ζ − ζ k k . Notice that there exists a constant C > i∂ ¯ ∂ (cid:0) χ ( ζ ) log k ζ k (cid:1) > − C i∂ ¯ ∂ k ζ k . Let K be any compact subset in A . Observe that for each δ > k δ such that for every k > k δ one has that B ( ζ k ) ∩ K = ∅ (here B ( ζ k ) denote the ball of radius 1 with center at ζ k ) and, moreover, that the function u δ,k is a negative and plurisubharmonic on A . We can insure plurisubharmonicityof u δ,k ( ζ ) for large enough k δ in the following way: • On the set A \ B ( ζ k ) plurisubharmonicity of u δ,k ( ζ ) is clear, since on thisset χ ( ζ − ζ k ) ≡ • On the set B ( ζ k ) one has i∂ ¯ ∂u δ,k > δ · i∂ ¯ ∂ ˜ ϕ − C i∂ ¯ ∂ k ζ k > (cid:0) δ · ˜ ρ ′ ( k ζ k ) − C (cid:1) i∂ ¯ ∂ k ζ k . This, in view of our assumptions that lim t →∞ ˜ ρ ′ ( t ) = ∞ and that ζ k → ∞ as k → ∞ , implies plurisubharmonicity of u δ,k ( ζ ) on B ( ζ k ) for all largeenough k δ .Observe now that, since u δ,k is a negative plurisubharmonic function with (at least)a logarithmic pole at ζ k , it is included into the family of functions which definesthe pluricomplex Green function g A ( ζ, ζ k ). By the definition of the pluricomplexGreen function, for large enough k we have g A ( ζ, ζ k ) > u δ,k ( ζ ) = δ · ˜ ϕ for ζ ∈ K and, hence, also K ∩ A A ( ζ k , a ) ⊂ { ζ ∈ K : ˜ ϕ ≤ − a/δ } . KOBAYASHI AND BERGMAN COMPLETE DOMAIN 9
Since for every ε > δ > { ζ ∈ K : ˜ ϕ < − a/δ } is less than ε , we conclude by the above arguments, that thereexists k δ such that Z K ∩ A A ( ζ k ,a ) dV < ε for all k > k δ . By Theorem 4.1, the Bergman metric is complete.5. Kobayashi completeness
Let Ω be an arbitrary domain in C n . Recall that the Kobayashi pseudometric κ Ω at a point ( ζ, v ) ∈ Ω × C n is defined as κ Ω ( ζ, v ) := inf { r : ∃ h ∈ O (∆ , Ω) with h (0) = ζ and h ′ (0) = rv } , (14)where ∆ := { z ∈ C : | z | < } is the unit disc in C and O (∆ , Ω) denotes the set ofall holomorphic maps from ∆ to Ω.For any two point ζ , ζ ∈ Ω, the Kobayashi pseudodistance is defined by κ Ω ( ζ , ζ ) = inf (cid:26)Z κ Ω ( γ ( t ) , γ ′ ( t )) dt (cid:27) , (15)where the infimum is taken over all piecewise C curve γ : [0 , → Ω connecting ζ and ζ .The domain Ω is called Kobayshi hyperbolic if κ Ω is a distance. Ω is said tobe Kobayashi complete (abbr. κ -complete) if any κ Ω -Cauchy sequence { ζ j } j ∈ N converges to a point ζ ∈ Ω with respect to the Euclidean topology of Ω.When Ω is a bounded domain in C n , it is well-known that Ω is κ -complete if andonly if Ω is locally κ -complete, i.e., for any boundary point a ∈ ∂ Ω, there exists abounded neighborhood U of a such that every connected component of Ω ∩ U is κ -complete (see, for example, [23, Theorem 7.5.5]).For unbounded domains which are (possibly) not biholomorphic to boundeddomains in C n , Nikolov and Pflug gave a criterion of the κ -completeness, by intro-ducing the following notions of κ -points and κ ′ -points. Definition 5.1.
A point a ∈ ∂ Ω is called a κ -point for Ω if lim ζ → a κ Ω ( ζ, ξ ) = ∞ forany fixed ξ ∈ Ω . Definition 5.2.
A point a ∈ ∂ Ω is called a κ ′ -point for Ω if there is no κ Ω -Cauchysequence converging to a . It is clear that any κ -point is a κ ′ -point. Nikolov and Pflug (see [25, Proposition3.6]) have proved the following theorem for the κ -completeness. Theorem 5.1.
Let Ω be an open set in C n . Assume that ∞ is a κ ′ -point if Ω isunbounded. Then the following conditions are equivalent:(1) Ω is κ -complete.(2) Any finite boundary point of Ω admits a neighborhood U such that Ω ∩ U is κ -complete.(3) Any finite boundary point of Ω is a κ ′ -point.(4) Any boundary point of Ω is a κ -point. We will now use the above theorem to prove that the domain A is κ -complete.Observe first that, by our definition of the domain A (which is given in Section2), one has that A ⊂ U − = F d with d = e − . By Property ( F ) stated in Remark 3 of [17] and restated in Section2 above, we know that the domain U − = F e − , and hence also a smaller domain A , contains only discs of uniformly bounded size, say r . This implies that A isKobayashi hyperbolic. Since the domain A is strictly pseudoconvex at each finiteboundary point, it follows that A is κ -complete at these points (see, for example,the arguments of Lemma 2.1.1 and Lemma 2.1.3 in [15]). Hence, for proving that A is κ -complete we only need to check that ∞ is a κ ′ -point of A .Now suppose that { ζ j } is a κ -Cauchy sequence converging to ∞ . Then, bythe definitions (14), (15) of the Kobayashi pseudodistance, and in view of theboundedness by r of the size of holomorphic disks contained in A , we see that1 r d E ( ζ k , ζ l ) ≤ κ A ( ζ k , ζ l )for all k, l ∈ N , where d E ( ζ, ξ ) denotes the Euclidean distance between the points ζ, ξ ∈ C . The last inequality obviously implies that { ζ j } is also a Cauchy sequencewith respect to the Euclidean distance, and hence can not converge to ∞ . Thisshows that ∞ is a κ ′ -point of A , and, therefore, proves the Kobayashi completenessof A .6. Nonexistence of nonconstant bounded holomorphic functions
In this section we will show that the only bounded holomorphic functions definedin A are constants. For proving this we will need the following version of theLiouville type theorem which was proved in [21, Theorem 2.2] for a slightly differentWermer type set. The argument provided there can also be applied to the set E considered in our paper. But, since for this set there is a bit simpler proof, we willpresent it here for the reader convenience. Theorem 6.1.
Let φ be a continuous plurisubharmonic function defined on anopen neighborhood V ⊂ C of the Wermer type set E . If φ is bounded from above,then φ ≡ C on E for some constant C ∈ R .Proof. By construction of the Wermer type set (see Section 1), E is locally thelimit in the Hausdorff distance of analytic sets E n and, therefore, the complementof E is pseudoconvex. Due to a theorem of Slodkowski, we know that E is ananalytic multivalued function (see the definition of analytic multivalued functionsand Slodkowski’s theorem in [3, pages 15-16]).Now, for a given set F ⊂ C z,w and any point z ∈ C z , let us define the verticalslice F ( z ) of the set F by F ( z ) := { w ∈ C w : ( z , w ) ∈ F } . Then for any function φ , which is continuous and plurisubharmonic in a neighbour-hood of the set E , we can define the function˜ φ ( z ) := max w ∈E ( z ) φ ( z , w ) . KOBAYASHI AND BERGMAN COMPLETE DOMAIN 11
It follows from part (ii) of the definition of analytic multivalued functions that ˜ φ is a subharmonic function in the complex plane C z . If the function φ is furtherassumed to be bounded from above (without loss of generality we may assume thaton the set E one has φ φ = 0), the standard Liouville theorem assertsthat ˜ φ ≡ C z .The rest of the proof will be devoted to showing that even for the initial function φ one also has that φ ≡
0. In order to do this we first define the set A := { p ∈ E : φ ( p ) = 0 , p z := π z ( p ) / ∈ Z + i Z } , where π z : C z,w → C z is the canonical projection. Then we observe that for everypoint p = ( p , p ) ∈ A and every analytic disc D r ( p ) := { ( z, f ( z )) : z ∈ ∆ r ( p ) } ⊂ E around p such that ∆ r ( p ) ⊂ C z \ { Z + i Z } , p = ( p , f ( p )) and f : ∆ r ( p ) → C w holomorphic, the function φ ( z, f ( z )) is subharmonic on ∆ r ( p ) ⊂ C z . Here ∆ r ( p )denote the disc of radius r centered at p . Since φ | E φ ( p ) = 0, we concludefrom the maximum principle that φ ( z, f ( z )) is identically equal to zero on ∆ r ( p )and, therefore, the set A is open in the topology defined along the leaves of E .More precisely, we say that a curve η : [0 , → E follows the leaves of E if for each t ∈ [0 ,
1] there exist ε > r > f : ∆ r ( π z ( η ( t ))) → C w such that Γ( f ) := { ( z, f ( z )) ∈ C z,w : z ∈ ∆ r ( π z ( η ( t ))) } ⊂ E and for each t ′ ∈ ( t − ε, t + ε ) (or, t ′ ∈ [0 , t + ε ) for t = 0 and t ′ ∈ (1 − ε,
1] for t = 1, respectively)one has that η ( t ′ ) ∈ Γ( f ). Claim 1. If η : [0 , → E is a continuous curve which follows the leaves of E andsuch that φ ( η (0)) ∈ A and π z ( η ( t )) / ∈ Z + i Z for all t ∈ [0 , φ ( η (1)) ∈ A .The claim easily follows if we cover the curve η ([0 , D , D , ..., D l ⊂ E and apply the maximum principle to the restrictionof φ to each of these discs.Now, in analogy to the definitions (1) and (2) of the sets E n and E , respectively,for each m ∈ N and each n ≥ m we can consider the set E m,n := (cid:8) ( z, w ) ∈ C : w = ε m √ z − a m + ... + ε n √ z − a n (cid:9) and then define the set E m = lim n →∞ E m,n = n b ∈ C z,w : ∃ b n ∈ E m,n , n = m, m + 1 , ..., with b = lim n →∞ b n o . It is easy to see from these definitions that for each m ∈ N we have that E = E m ⊕ E m +1 := { ( z, w + w ) ∈ C z,w : w ∈ E m ( z ) , w ∈ E m +1 ( z ) } , (16)and then, from the construction of the set E (more precisely, from the choice of thesequence { ε n } in this construction) we also conclude that, if for every compact set K ⊂ C z and every m ∈ N we define δ K ( m ) := max z ∈ K max w ∈E m ( z ) | w | , then lim m →∞ δ K ( m ) = 0 . (17)Next, for a fixed m ∈ N , we define the notion of the lifting ˜ γ of a curve γ ⊂ C z tothe set E m . Let p be a point of E m such that π z ( p ) / ∈ Z + i Z . Then for some r > r ( π z ( p )) ⊂ C z \ { Z + i Z } and, hence, the set E m ∩ (∆ r ( π z ( p )) × C w )can be represented as the union of the graphs { Γ( f α ) } α ∈B of holomorphic functions { f α : ∆ r ( π z ( p )) → C w } α ∈B . Let f α ( p ) be a function of this family such that itsgraph Γ( f α ( ζ ) ) contains the point p and let γ : [0 , → C z \{ Z + i Z } be a continuouscurve such that π z ( p ) = γ (0). Then we define the lifting of the curve γ with theinitial data f α ( p ) as follows:We divide the segment [0 ,
1] into small enough segments by points 0 = t < t <... < t k = 1 and then consider a family of disks { ∆ r s ( γ ( t s )) } ≤ s ≤ k such that foreach s = 1 , , ..., k one has ∆ r s ( γ ( t s )) ⊂ C z \ { Z + i Z } and γ ([ t s − , t s ]) ⊂ ∆ r s ( γ ( t s ))(in particular, { ∆ r s ( γ ( t s ) } ≤ s ≤ k is a covering of γ ([0 , s = 1 , , ..., k the set E m ∩ (∆ r s ( γ ( t s )) × C w ) can be represented as the unionof the graphs { Γ( f sα ) } α ∈B of holomorphic functions { f sα : ∆ r s ( γ ( t s )) → C w } α ∈B .Note now that, in view of the construction of the set E m and the unicity theoremfor holomorphic functions, there exists exactly one function f α ( p ) in the family { f α :∆ r ( γ ( t )) → C w } α ∈B which coincides with f α ( p ) on the set ∆ r ( γ ( t )) ∩ ∆ r ( γ ( t ))(without loss of generality we can assume here that r < r , and hence the function f α ( p ) is defined on the disk ∆ r ( γ ( t ))). Then we proceed inductively and, if forsome 2 ≤ s ≤ k the function { f s − α ( p ) : ∆ r s − ( γ ( t s − )) → C w } is already chosen,we consider the (uniquely defined) function from the family { f sα : ∆ r s ( γ ( t s )) → C w } α ∈B which coincides with f s − α ( p ) on the set ∆ r s − ( γ ( t s − )) ∩ ∆ r s ( γ ( t s )) anddenote it by f sα ( p ) . Now we can finally define the lifting ˜ γ of the curve γ to the set E m . Namely, for each s = 1 , , ..., k and each θ ∈ [ t s − , t s ] we set˜ γ ( θ ) := ( γ ( θ ) , f sα ( p ) ( γ ( θ ))) . To finish the proof of the theorem we need to show that for an arbitrary point q ∈ E one has φ ( q ) = 0. In view of continuity of the function φ , it is enough to provethis claim for points q such that π z ( q ) / ∈ Z + i Z . Let us fix a point q with theseproperties and let p be a point of the set A , i.e. φ ( p ) = 0 and π z ( p ) / ∈ Z + i Z . Fix R > π z ( p ) , π z ( q ) ∈ ∆ R (0) and set K := ∆ R (0). Then, by property(17) above, for each n ∈ N there is m n ∈ N such thatmax z ∈ K max w ∈E m ( z ) | w | < n (18)for all m ≥ m n . Further, in view of property (16), we can find points p n , q n ∈ E m n such that π z ( p n ) = π z ( p ), π z ( q n ) = π z ( q ) and˜ p n := ( π z ( p n ) , π w ( p ) − π w ( p n )) ∈ E m n +1 , (19) ˜ q n := ( π z ( q n ) , π w ( q ) − π w ( q n )) ∈ E m n +1 , (20)where π w : C z,w → C w is the canonial projection. Since the set E m n \ ( { Z + i Z }× C w )is obviously connected, there is a continuous curve η n : [0 , → E m n such that η n (0) = p n , η n (1) = q n and π z ( η n ( t )) / ∈ Z + i Z for all t ∈ [0 , γ n : [0 , → C z \ { Z + i Z } be a curve defined by γ n ( t ) := π z ( η n ( t )). Consider now some initialdata f α (˜ p n ) for the lifting of the curve γ n to the set E m n +1 , that is, consider anumber r > f α (˜ p n ) : ∆ r ( π z ( p n )) → C w such thatits graph Γ( f α (˜ p n ) ) contains the point ˜ p n and is contained in the set E m n +1 . Inthe case when such initial data is not unique, one can just choose an arbitraryone. Then, in view of the construction above of the lifting of a curve, we will geta lifting ˜ γ n of the curve γ n to the set E m n +1 with the initial point ˜ γ n (0) = ˜ p n KOBAYASHI AND BERGMAN COMPLETE DOMAIN 13 and an endpoint ˜ γ n (1) =: ˜˜ q n ∈ E m n +1 ( π z ( q )). Now we can finally define the curve γ ∗ n : [0 , → E \ ( { Z + i Z } × C w ) by γ ∗ n ( t ) := ( γ n ( t ) , π w ( η n ( t )) + π w (˜ γ n ( t )))and observe that, by our construction and property (19), one has that γ ∗ n (0) = ( π z ( η n (0)) , π w ( η n (0)) + π w (˜ γ n (0))) = ( π z ( p n ) , π w ( p n ) + π w (˜ p n ))= ( π z ( p ) , π w ( p )) = p (21)and, by property (20), one also has that γ ∗ n (1) = ( π z ( η n (1)) , π w ( η n (1)) + π w (˜ γ n (1))) = ( π z ( q n ) , π w ( q n ) + π w (˜˜ q n )= ( π z ( q ) , π w ( q ) − π w (˜ q n ) + π w (˜˜ q n )) =: q ∗ n . (22)Since, by properties (21) and (22), γ ∗ n is a curve in E \ ( { Z + i Z } × C w ) connectingthe points p and q ∗ n , and since, by the choice of p , one has that φ ( p ) = 0, we canconclude from Claim 1 that φ ( q ∗ n ) = 0. But properties (18), (20), (22) and the factthat ˜˜ q n ∈ E m n +1 ( π z ( q )) imply that k q − q ∗ n k = | π w (˜ q n ) − π w (˜˜ q n ) | < | π w (˜ q n ) | + | π w (˜˜ q n ) | < n − . Hence, by continuity of φ , we finally have that φ ( q ) = lim n →∞ φ ( q ∗ n ) = 0. Thisconcludes the proof of our Liouville type theorem. (cid:3) Let us now consider an arbitrary bounded holomorphic function f on the domain A . Without loss of generality, we can assume that | f | < A . Then the followingstatement holds true. Claim 2.
The restriction f | E of the function f to the Wermer type set E ⊂ A isconstant. Proof.
Due to the last theorem, we know that | f | ≡ r on E for some constant r <
1. Hence, we can write f ( z, w ) = r exp( iθ ) , where θ ∈ [0 , π ], in principle, mightbe different for different values z and w . Consider now the function g := | − f | .Then g is also bounded continuous and plurisubharmonic on some neighborhoodof E , hence, by the last theorem, | g | is also constant (say C ∗ ) on E . A simplecalculation shows that there are at most two solutions θ and 2 π − θ to the equation | − r exp( iθ ) | = C ∗ . It follows then from holomorphicity (and, hence, continuity)of f and connectedness of E that the restriction f | E of f to E is constant. Thiscompletes the proof of Claim 2. (cid:3) Now we fix z = z and consider the slice A z := A ∩ { z = z } . Since, byconstruction, the core E intersected with A z is a Cantor set, it follows that there isa point w and a sequence of points w j converging to w such that ζ j = ( z , w j ) ∈E ∩ A z for each j ∈ N . Then, by Claim 2, there is a constant C such that f | E ≡ C .In particular, we have that f ( ζ j ) = C for each j ∈ N . The standard one-dimensionalidentity theorem for holomorphic functions tells us now that f = C on the wholeslice A z . Since the same argument holds true for each z , we finally conclude that f ≡ C on A , which completes the proof of the main statement of this Section.The next statement follows directly from nonexistence of bounded holomorphicfunctions. Corollary 6.1.
The domain A has the following properties:(1) A can not be biholomorphic to a bounded domain.(2) The Carath´eodory metric on A is identically equal to zero. Acknowledgement.
Part of this work was done while the first author was avisitor at the Capital Normal University (Beijing). It is his pleasure to thank thisinstitution for its hospitality and good working conditions.
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Nikolay Shcherbina: Department of Mathematics, University of Wuppertal, 42119,Wuppertal, Germany
E-mail address : [email protected] Liyou Zhang: School of Mathematical Sciences, Capital Normal University, 100048,Beijing, P. R. China
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